《数论小火花》

小草

施承忠素数个数π(x)分段系数法

(pk)^2≤x<(pk+1)^2；π(x)≈bk*x/lnx

2
ln4=1.3862943611
4/1.3862943611=2.8853900818
3/2.8853900818=1.0397207708
b1=1.0397207708

3
ln9=2.1972245773
9/2.1972245773=4.0960765198
6/4.0960765198=1.4648163849
b2=1.4648163849

5
ln25=3.2188758248
25/3.2188758248=7.7666866822
11/7.7666866822=1.4163053629
b3=1.4163053629

7
ln49=3.8918202981
49/3.8918202981=12.5905093881  
18/12.5905093881=1.4296482728  
b4=1.4296482728  

11
ln121=4.7957905456
121/4.7957905456=25.2304596812
29/25.2304596812=1.1494043456
b5=1.1494043456

13
ln169=5.1298987149
169/5.1298987149=32.9441202239
42/32.9441202239=1.2748860712
b6=1.2748860712

17
ln289=5.6664266881
289/5.6664266881=51.0021598986
59/51.0021598986=1.1568137529
b7=1.1568137529

19
ln361=5.8888779583
361/5.8888779583=61.3020005774
78/61.3020005774=1.2723891433
b8=1.2723891433

23
ln529=6.2709884319
529/6.2709884319=84.3567175645
101/84.3567175645=1.1972964681
b9=1.1972964681

29
ln841=6.7345916600
841/6.7345916600=124.8776529385
130/124.8776529385=1.0410189249
b10=1.0410189249

31
ln961=6.8679744090
961/6.8679744090=139.9248079231
161/139.9248079231=1.1506179811
b11=1.1506179811

37
ln1369=7.2218358253
1369/7.2218358253=189.5639880381
198/189.5639880381=1.0445021866
b12=1.0445021866

41
ln1681=7.4271441334
1681/7.4271441334=226.3319480284
239/226.3319480284=1.0559711171
b13=1.0559711171

43
ln1849=7.5224002314
1849/7.5224002314=245.7992054560
282/245.7992054560=1.1472779152
b14=1.1472779152

47
ln2209=7.7002952034
2209/7.7002952034=286.8721187500
329/286.8721187500=1.1468524771
b15=1.1468524771

53
ln2809=7.9405838271
2809/7.9405838271=353.75232 61720
382/353.75232 61720=1.07985155641
b16=1.07985155641

59
ln3481=8.1550748878
3481/8.1550748878=426.8507705806
441/426.8507705806=1.0331479533
b17=1.0331479533

61
ln3721=8.2217477283
3721/8.2217477283=452.5801718766
502/452.5801718766=1.1091957430
b18=1.1091957430

67
ln4489=8.4093852388
4489/8.4093852388=533.8083430033
569/533.8083430033=1.0659256406
b19=1.0659256406

71
ln5041=8.5253597541
5041/8.5253597541=591.2946955201
640/591.2946955201=1.0823706095
b20=1.0823706095

73
ln5329=8.5809188823
5329/8.5809188823=621.0290614671
713/621.0290614671=1.1480944198
b21=1.1480944198

79
ln6241=8.7388957049
6241/8.7388957049=714.1634607792
792/714.1634607792=1.1089898090
b22=1.1089898090

83
ln6889=8.8376812156
6889/8.8376812156=779.5031108205
875/779.5031108205=1.1225099526
b23=1.1225099526

89
ln7921=8.9772727395
7921/8.9772727395=882.3392393046
964/882.3392393046=1.0925502993
b24=1.0925502993

97
ln9409=9.1494219570
9409/9.1494219570=1028.3709773382
1061/1028.3709773382=1.0317288443
b25=1.0317288443

101
ln10201=9.2302410337
10201/9.2302410337=1105.1715727418
1162/1105.1715727418=1.0514204569
b26=1.0514204569

小草

施承忠孪生素数对数T(x)分段系数法

2(qk)^2≤x<2(qk+1)^2；T(x)≈ck*x/ln^2x

3
(ln18)^2=8.3542488988
18/8.3542488988=2.1545922581
3/2.1545922581=1.3923748165
c1=1.3923748165

5
(ln50)^2=15.3039239950
50/15.3039239950=3.2671359330
8/3.2671359330=2.4486278392
c2=2.4486278392

11
(ln242)^2=30.1284373616
242/30.1284373616=8.0322785113
19/8.0322785113=2.3654558259
c3=2.3654558259

17
(ln578)^2=40.4441797911
578/40.4441797911=14.2913023082
36/14.2913023082=2.5190146583
c4=2.5190146583

29
(ln1682)^2=55.1713042832
1682/55.1713042832=30.4868630868
65/30.4868630868=2.1320658611
c5=2.1320658611

41
(ln3362)^2=65.9391310237
3362/65.9391310237=50.9864165300
106/50.9864165300=2.07898509474
c6=2.07898509474

59
(ln6962)^2=78.2910337712
6962/78.2910337712=88.9246145395
165/88.9246145395=1.8555042477
c7=1.8555042477

71
(ln10082)^2=84.9808701041
10082/84.9808701041=118.6384651940
236/118.6384651940=1.9892367928
c8=1.9892367928

101
(ln20402)^2=98.4736336506
20402/98.4736336506=207.1823618532
337/207.1823618532=1.6265863415
c9=1.6265863415

107
(ln22898)^2=100.7776028060
22898/100.7776028060=227.2131839063
444/227.2131839063=1.9541119594
c10=1.9541119594

137
(ln37538)^2=110.9463858846
37538/110.9463858846=338.3436035406
581/338.3436035406=1.7171892535
c11=1.7171892535

149
(ln44402)^2=114.5122526396
44402/114.5122526396=387.7489000216
730/387.7489000216=1.8826616915
c12=1.8826616915

179
(ln64082)^2=122.4988263917
64082/122.4988263917=523.1233791179
909/523.1233791179=1.7376397926
c13=1.7376397926

191
(ln72962)^2=125.3883517385
72962/125.3883517385=581.8881817042
1100/581.8881817042=1.8903975619
c14=1.8903975619

197
(ln77618)^2=126.7775706469
77618/126.7775706469=612.2376348115
1297/612.2376348115=2.1184584649
c15=2.1184584649

227
(ln103058)^2=133.2419390198
103058/133.2419390198=773.4651773920
1524/773.4651773920=1.97035 37335
c16=1.97035 37335

239
(ln114242)^2=135.6310462383
114242/135.6310462383=842.2997769941
1763/842.2997769941=2.0930790298
c17=2.0930790298

269
(ln144722)^2=141.1954683734
144722/141.1954683734=1024.9762380282
2032/1024.9762380282=1.9824849832
c18=1.9824849832

281
(ln157922)^2=143.2774650909
157922/143.2774650909=1102.2110134333
2313/1102.2110134333=2.0985092435
c19=2.0985092435

311
(ln193442)^2=175.5035992586
193442/175.5035992586=1102.2110134332
2624/1102.2110134332=2.3806693709
c20=2.3806693709

小草

施承忠哥德巴赫偶数素数对数D(x)分段系数法

4(qk)^2≤x<4(qk+1)^2；D(2n)≈dk*2n/ln^2(2n)

3
(ln36)^2=12.8416079823
36/12.8416079823=2.8033872432
3/2.8033872432=1.0701339985
d1=1.0701339985

5
(ln100)^2=35.6711332844
100/35.6711332844=2.8033872432
8/2.8033872432=2.8536906628
d2=2.8536906628

11
(ln484)^2=38.2181737939
484/38.2181737939=12.6641320595
19/12.6641320595=1.5003002109
d3=1.5003002109

17
(ln1156)^2=49.7408741983
1156/49.7408741983=23.2404439735
36/23.2404439735=1.5490237640
d4=1.5490237640

29
(ln3364)^2=65.9487897676
3364/65.9487897676=51.0092757101
65/51.0092757101=1.2742780425
d5=1.2742780425

41
(ln6724)^2=77.6766980968
6724/77.6766980968=86.5639266955
106/86.5639266955=1.2245285542
d6=1.2245285542

59
(ln13924)^2=91.0377271445
13924/91.0377271445=152.9475793909
165/152.9475793909=1.0788009896
d7=1.0788009896

71
(ln20164)^2=98.2408872994
20164/98.2408872994=205.2505891824
236/205.2505891824=1.1498139954
d8=1.1498139954

101
(ln40804)^2=112.7108237891
40804/112.7108237891=362.0237935298
337/362.0237935298=0.9308780418
d9=0.9308780418

107
(ln45796)^2=115.1747943752
45796/115.1747943752=397.6217213882
444/397.6217213882=1.1166391978
d10=1.1166391978

137
(ln75076)^2=126.0288285549
75076/126.0288285549=595.7049737021
581/595.7049737021=0.9753150060
d11=0.9753150060

149
(ln88804)^2=129.8274967760
88804/129.8274967760=684.0153450175
730/684.0153450175=1.0672275196
d12=1.0672275196

179
(ln128164)^2=138.3226728166
128164/138.3226728166=926.5581512434
909/926.5581512434=0.9810501357
d13=0.9810501357

191
(ln145924)^2=141.3921048530
145924/141.3921048530=1032.0519674823
1100/1032.0519674823=1.0658378014
d14=1.0658378014

197
(ln155236)^2=142.8670807643
155236/142.8670807643=1086.5764119315
1297/1086.5764119315= 1.1936574232
d15=1.1936574232

227
(ln206116)^2=149.7244532987
206116/149.7244532987=1376.6355158352
1524/1376.6355158352=1.1070468417
d16=1.1070468417

239
(ln228484)^2=152.2563863619
228484/152.2563863619=1500.6529805385
1763/1500.6529805385=1.1748219094
d17=1.1748219094

269
(ln289444)^2=158.1486610904
289444/158.1486610904=1830.2020263994
2032/1830.2020263994=1.1102599444
d18=1.1102599444

281
(ln315844)^2=160.3516627008
315844/160.3516627008=1969.6958215478
2313/1969.6958215478=1.1742929922
d19=1.1742929922

311
(ln386884)^2=165.5308729471
386884/165.5308729471=2337.2316783688
2624/2337.2316783688=1.1226957192
d20=1.1226957192

小草

103
ln10609=9.2694579765
10609/9.2694579765=1144.5113648388
1265/1144.5113648388=1.1052751758
b27=1.1052751758

107
ln11449=9.3456576689
11449/9.3456576689=1225.0609219402
1372/1225.0609219402=1.1199443027
b28=1.1199443027

109
ln11881=9.3826957645
11881/9.3826957645=1266.2672112798
1481/1266.2672112798=1.1695793643
b29=1.1695793643

113
ln12769=9.4547756374
12769/9.4547756374=1350.5344272253
1594/1350.5344272253=1.1802735035
b30=1.1802735035

127
ln16129=9.6883741729
16129/9.6883741729=1664.7788072756
1721/1664.7788072756=1.0337709685
b31=1.0337709685

131
ln17161=9.7503946464
17161/9.7503946464=1760.0313240999
1852/1760.0313240999=1.0522539995
b32=1.0522539995

137
ln18769=9.8399618517
18769/9.8399618517=1907.4260940105
1989/1907.4260940105=1.0427664832
b33=1.0427664832

139
ln19321=9.8689478663
19321/9.8689478663=1957.7568208640
2128/1957.7568208640=1.0869582868
b34=1.0869582868

149
ln22201=10.0078926119
22201/10.0078926119=2218.3491431155
2277/2218.3491431155=1.0264389657
b35=1.0264389657

151
ln22801=10.0345596736
22801/10.0345596736=2272.2471878848
2428/2272.2471878848=1.0685457167
b36=1.0685457167

157
ln24649=10.1124916107
24649/10.1124916107=2437.4803904825
2585/2437.4803904825=1.0605213523
b37=1.0605213523

163
ln26569=10.1875004016
26569/10.1875004016=2607.9998971904
2748/2607.9998971904=1.0536810231
b38=1.0536810231

167
ln27889=10.2359876248
27889/10.2359876248=2724.6027469230
2915/2724.6027469230=1.0698807389
b39=1.0698807389

173
ln29929=10.3065831890
29929/10.3065831890=2903.8721612360
3088/2903.8721612360=1.0634076945
b40=1.0634076945

179
ln32041=10.3747716117
32041/10.3747716117=3088.3571416518
3267/3088.3571416518=1.0578439766
b41=1.0578439766

181
ln32761=10.3969940625
32761/10.3969940625=3151.0068970956
3448/3151.0068970956=1.0942533966
b42=1.0942533966

191
ln36481=10.5045468561
36481/10.5045468561=3472.8770788257
3639/3472.8770788257=1.0478343798
b43=1.0478343798

193
ln37249=10.5253803778
37249/10.5253803778=3538.9694873703
3832/3538.9694873703=1.0828010848
b44=1.0828010848

197
ln38809=10.5664074575
38809/10.5664074575=3672.86612 37130
4029/3672.86612 37130=1.0969634787
b45=1.0969634787

199
ln39601=10.5866096494
39601/10.5866096494=3740.6687609611
4228/3740.6687609611=1.1302791747
b46=1.1302791747

211
ln44521=10.7037162670
44521/10.7037162670=4159.3965020598
4439/4159.3965020598=1.06722 21313
b47=1.06722 21313

223
ln49729=10.8143435429
49729/10.8143435429=4598.4298355907
4662/4598.4298355907=1.0138243198
b48=1.0138243198

227
ln51529=10.8499000350
51529/10.8499000350=4749.2603465263
4889/4749.2603465263=1.0294234561
b49=1.0294234561

229
ln52441=10.8674440071
52441/10.8674440071=4825.5137054986
5351/4825.5137054986=1.1088974825
b50=1.1088974825

233
ln54289=10.9020769071
54289/10.9020769071=4979.6933614222
5351/4979.6933614222=1.0745641572
b51=1.0745641572

239
ln57121=10.9529271039
57121/10.9529271039=5215.1355941793
5590/5215.1355941793=1.0718800881
b52=1.0718800881

241
ln58081=10.9695938670
58081/10.9695938670=5294.72655999834
5831/5294.7265599983=1.10128444480
b53=1.10128444480

251
ln63001=11.0509058783
63001/11.0509058783=5700.9805977726
6082/5700.9805977726=1.0668340114
b54=1.0668340114

257
ln66049=11.0981521698
66049/11.0981521698=5951.3510888534
6339/5951.3510888534=1.0651362868
b55=1.0651362868

263
ln69169=11.1443080644
69169/11.1443080644=6206.6661833369
6602/6206.6661833369=1.0636950345
b56=1.0636950345

269
ln72361=11.1894227592
72361/11.1894227592=6466.9108994478
6871/6466.9108994478=1.0624856453
b57=1.0624856453

271
ln73441=11.2042376418
73441/11.2042376418=6554.7520811243
7142/6554.7520811243=1.0895911717
b58=1.0895911717

277
ln76729=11.2480350124
76729/11.2480350124=6821.5470449205
7419/6821.5470449205=1.0875832053
b59=1.0875832053

281
ln78961=11.2767093387
78961/11.2767093387=7002.1313512992
7700/7002.1313512992=1.0996651753
b60=1.0996651753

283
ln80089=11.2908937953
80089/11.2908937953=7093.2382725394
7983/7093.2382725394=1.1254380148
b61=1.1254380148

293
ln85849=11.3603452180
85849/11.3603452180=7556.9006357286
8276/7556.9006357286=1.0951579753
b62=1.0951579753

307
ln94249=11.4536954952
94249/11.4536954952=8228.6978940114
8583/8228.6978940114=1.0430568859
b63=1.0430568859

311
ln96721=11.4795858244
96721/11.4795858244=8425.4781905475
8894/8425.4781905475=1.0556077411
b64=1.0556077411

313
ln97969=11.4924063811
97969/11.4924063811=8524.6724446776
9207/8524.6724446776=1.0800414983
b65=1.0800414983

317
ln100489=11.5178035478
100489/11.5178035478=8724.6669543339
9524/8724.6669543339=1.0916175998
b66=1.0916175998

331
ln109561=11.6042367508
109561/11.6042367508=9441.4654192958
9855/9441.4654192958=1.0437998300
b67=1.0437998300

337
ln113569=11.6401658607
113569/11.6401658607=9756.6479171432
10192/9756.6479171432=1.0446210714
b68=1.0446210714

347
ln120409=11.6986495599
120409/11.6986495599=10292.5555110849
10539/10292.5555110849=1.0239439553
b69=1.0239439553

349
ln121801=11.7101438444
121801/11.7101438444=10401.3239818781
10888/10401.3239818781=1.0467898144
b70=1.0467898144

小草

D(6)=1
1*2=2
ln6=1.79176
(1.79176)^2=3.21040
6/3.21040=1.86893
1.86893*1.32032=2.46759
2.46759*2=4.93518
【】
D(36)=4
4*2=8
ln36=3.58352
(3.58352)^2=12.84162
36/12.84162=2.80338
2.80338*1.32032=3.70135
3.70135*2=7.4027
【】
D(30)=3
3*2=6
ln30=3.40120
(3.40120)^2=11.56816
30/11.56816=2.59333
2.59333*1.32032=3.42403
3.42403*2.66667=9.13075
【】
D(900)=48
48*2=96
(ln900)^2=46.27257
900/46.27257=19.44997
19.44997*1.32032=25.68018
25.68018*2.66667=68.48057
【】
D(210)=19
19*2=38
(ln210)^2=28.59156
210/28.59156=7.34482
7.34482*1.32032=9.69751
9.69751*3.2=31.03203
【】
D(44100)=1007
1007*2=2014
(ln44100)^2=114.36624
44100/114.36624=385.60330
385.60330*1.32032=509.11975
509.11975*3.2=1629.1832
【】
D(2310)=114
114*2=228
(ln2310)^2=59.98507
2310/59.98507=38.50958
38.50958*1.32032=50.84497
50.84497*3.55556=180.78234
【】
D(36960)=980
980*2=1960
(ln36960)^2=110.61973
36960/110.61973=334.11761
334.11761*1.32032=441.14216
441.14216*3.55556=1568.50742
【】
D(30030)=905
905*2=1810
(ln30030)^2=106.29511
30030/106.29511=282.51535
282.51535*1.32032=373.01067
373.01067*3.87878=492.49345

小草

5D(30)=3
7D(210)=19
11D(2310)=114
13D(30030)=905
17D(510510)=9493
19D(9699690)=124180
23D(223092870)=2044847
29D(6469693230)≈44128292
31D(200560490130)≈1015462775

小草

哥德巴赫偶数素数对的真值上限中值与下限
【4(q1)^2】
D(48)=5
(ln48)^2=14.9861972669
48/14.9861972669=3.2029472951
5/3.2029472951=1.5610622153

D(36)=4
(ln36)^2=12.8416079823
36/12.8416079823=2.8033872432
4/2.8033872432=1.4268453314

D(68)=2
(ln68)^2=17.8042452740
68/17.8042452740=3.8193138183
2/3.8193138183=0.5236542728
【4(q2)^2】
D(114)=10
(ln114)^2=22.4315757426
114/22.4315757426=5.0821217960
10/5.0821217960=1.9676820827

D(100)=6
(ln100)^2=21.2075924419
100/21.2075924419=4.7152924253
6/4.7152924253=1.2724555465

D(128)=3
(ln128)^2=23.5421976820
128/23.5421976820=5.4370455014
3/5.4370455014=0.5517702582
【4(q3)^2】
D(510)=32
(ln510)^2=38.8678770970
510/38.8678770970=13.1213752356
32/13.1213752356=2.4387687590

D(484)=14
(ln484)^2=38.2181737939
484/38.2181737939=12.6641320595
14/12.6641320595=1.1054843659

D(488)=9
(ln488)^2=38.3200048239
488/38.3200048239=12.7348626975
9/12.7348626975=0.7067214004
【4(q4)^2】
D(1260)=68
(ln1260)^2=50.9634220429
1260/50.9634220429=24.7236144963
68/24.7236144963=2.7504069039

D(1156)=22
(ln1156)^2=49.7408741983
1156/49.7408741983=23.2404439735
22/23.2404439735=0.9466256335

D(1412)=18
(ln1412)^2=52.6025626927
1412/52.6025626927=26.8427986722
18/26.8427986722=0.6705709125
【4(q5)^2】
D(3570)=154
(ln3570)^2=66.9176496143
3570/66.9176496143=53.3491540809
154/53.3491540809=2.8866437088

D(3364)=47
(ln3364)^2=65.9487897676
3364/65.9487897676=51.0092757100
47/51.0092757100=0.9214010461

D(3632)=40
(ln3632)^2=67.1996413892
3632/67.1996413892=54.0479074727
40/54.0479074727=0.7400841563
【4(q6)^2】
D(6930)=268
(ln6930)^2=78.2095278987
6930/78.2095278987=88.6081298045
268/88.6081298045=3.0245531713

D(6724)=71
(ln6724)^2=77.6766980968
6724/77.6766980968=86.5639266955
71/86.5639266955=0.8202030882

D(7292)=67
(ln7292)^2=79.1127197038
7292/79.1127197038=92.1722831335
67/92.1722831335=0.7268996462
【4(q7)^2】
D(15330)=447
(ln15330)^2=92.8826971372
15330/92.8826971372=165.0468867991
447/165.0468867991=2.7083213060

D(13924)=131
(ln13924)^2=91.0377271445
13924/91.0377271445=152.9475793909
131/152.9475793909=0.8565026038

D(14138)=117
(ln14138)^2=91.3290142406
14138/91.3290142406=154.8029409663
117/154.8029409663=0.7557995944
【4(q8)^2】
D(20790)=615
(ln20790)^2=98.8478852926
20790/98.8478852926=210.3231641067
615/210.3231641067=2.9240716429

D(20164)=175
(ln20164)^2=98.2408872994
20164/98.2408872994=205.2505891824
175/205.2505891824=0.8526163101

D(20348)=155
(ln20348)^2=98.4210405984
20348/98.4210405984=206.7444103038
155/206.7444103038=0.7497179719
【4(q9)^2】
D(43680)=1083
(ln43680)^2=114.1616518202
43680/114.1616518202=382.6153467786
1083/382.6153467786=2.8305189771

D(40804)=309
(ln40804)^2=112.7108237891
40804/112.7108237891=362.0237935298
309/362.0237935298=0.8535350591

D(40814)=280
(ln40814)^2=112.7160268855
40814/112.7160268855=362.0958006394
280/362.0958006394=0.7732760212
【4(q10)^2】
D(46410)=1205
(ln46410)^2=115.4608323638
46410/115.4608323638=401.9544901060
1205/401.9544901060=2.9978518207

D(45796)=333
(ln45796)^2=115.1747943752
45796/115.1747943752=397.6217213882
333/397.6217213882=0.8374793984

D(45998)=296
(ln45998)^2=115.2692799550
45998/115.2692799550=399.0482114398
296/399.0482114398=0.7417650086
【4(q11)^2】
D(76230)=1692
(ln76230)^2=126.3715552860
76230/126.3715552860=603.2211903025
1692/603.2211903025=2.8049412507

D(75076)=483
(ln75076)^2=126.0288285549
75076/126.0288285549=595.7049737021
483/595.7049737021=0.8108040411

D(75188)=447
(ln75188)^2=126.0623009610
75188/126.0623009611=596.4352500848
447/596.4352500848=0.7494526857
【4(q12)^2】
D(90090)=2135
(ln90090)^2=130.1553428017
90090/130.1553428017=692.1728917211
2135/692.1728917211=3.0844894759

D(88804)=549
(ln88804)^2=129.8274967759
88804/129.8274967759=684.0153450181
549/684.0153450181=0.8026135729

D(89372)=521
(ln89372)^2=129.9728301730
89372/129.9728301730=687.6206348745
521/687.6206348745=0.7576852316
【4(q13)^2】
D(131670)=2810
(ln131670)^2=138.95821878901
131670/138.95821878901=947.5510059605
2810/947.5510059605=2.9655395671

D(128164)=755
(ln128164)^2=138.3226728166
128164/138.3226728166=926.5581512434
755/926.5581512434=0.8148436221

D(129524)=704
(ln129524)^2=138.5710720597
129524/138.5710720597=934.711683144
704/934.711683144=0.7531734252
【4(q14)^2】
D(150150)=3215
(ln150150)^2=142.0718597221
150150/142.0718597221=1056.8595377980
3215/1056.8595377980=3.0420314952

D(145924)=841
(ln145924)^2=141.3921048530
145924/141.3921048530=1032.0519674823
841/1032.0519674823=0.8148814464

D(147248)=790
(ln147248)^2=142.6749859886
147248/142.6749859886=1032.0519674820
790/1032.0519674820=0.7654653301
【4(q15)^2】
D(157080)=3320
(ln157080)^2=143.1495117126
157080/143.1495117126=1097.3142564074
3320/1097.3142564074=3.0255690023

D(155236)=852
(ln155236)^2=142.8670807643
155236/142.8670807643=1086.5764119315
852/1086.5764119315=0.7841142055

D(158428)=826
(ln158428)^2=143.3540582555
158428/143.3540582555=1105.1518312627
826/1105.1518312627=0.7474086154
【4(q16)^2】
D(207480)=4033
(ln207480)^2=149.8859125782
207480/149.8859125782=1384.2528389167
4033/1384.2528389167=2.9134850850

D(206116)=1083
(ln206116)^2=149.7244532987
206116/149.7244532987=1376.6355158352
1083/1376.6355158352=0.7867006100

D(206498)=1042
(ln206498)^2=149.7697700476
206498/149.7697700476=1378.7695603350
1042/1378.7695603350=0.7557463045
【4(q17)^2】
D(232050)=4470
(ln232050)^2=152.6388133400
232050/152.6388133400=1520.2555295232
4470/1520.2555295232=2.9402951762

D(228484)=1195
(ln228484)^2=152.2563863619
228484/152.2563863619=1500.6529805385
1195/1500.6529805385=0.7963200124

D(230228)=1142
(ln230228)^2=152.4440974886
230228/152.4440974886=1510.2454197495
1142/1510.2454197495=0.7561684909
【4(q18)^2】5240/2032
D(297990)=5398
(ln297990)^2=158.8813664247
297990/158.8813664247=1875.5503348546
5398/1875.5503348546=2.8780885800

D(289444)=1439
(ln289444)^2=154.3248371537
289444/154.3248371537=1875.5503348546
1439/1875.5503348546=0.7672414721

D(291008)=1385
(ln291008)^2=158.2842291189
291008/158.2842291189=1838.5154454106
1385/1838.5154454106=0.7533251915
【4(q19)^2】
D(324870)=5926
(ln324870)^2=161.0660594705
324870/161.0660594705=2016.9984978089
5926/2016.9984978089=2.9380289606

D(315844)=1544
(ln315844)^2=156.5910933216
315844/156.5910933216=2016.9984978094
1544/2016.9984978094=0.7654938770

D(317972)=1494
(ln317972)^2=160.5217695950
317972/160.5217695950=1980.8652795334
1494/1980.8652795334=0.7542158548
【4(q20)^2】
D(390390)=7094
(ln390390)^2=165.7630891095
390390/165.7630891095=2355.1081371445
7094/2355.1081371445=3.0121759116

D(386884)=1830
(ln386884)^2=165.53087294708465
386884/165.53087294708465=2337.2316783690
1830/2337.2316783690=0.7829775785

D(388142)=1774
(ln388142)^2=165.6144176717
388142/165.6144176717=2343.6486113752
1774/2343.6486113752=0.7569394112
【4(q21)^2】
D(510510)=9493
(ln510510)^2=172.7427994927
510510/172.7427994927=2955.3185516226
9493/2955.3185516226=3.2121748753

D(481636)=2219
(ln481636)^2=171.2157574161
481636/171.2157574161=2813.0354779759
2219/2813.0354779759=0.7888275912

D(482072)=2119
(ln482072)^2=171.2394377557
482072/171.2394377557=2815.1926116912
2119/2815.1926116912=0.7527016060
【4(q22)^2】
D(746130)=12684
(ln746130)^2=182.8622016767
746130/182.8622016767=4080.2855546886
12684/4080.2855546886=3.10860596151781

D(702244)=2977
(ln702244)^2=181.2264186761
702244/181.2264186761=3874.9538016038
2977/3874.9538016038=0.7682672239

D(705542)=2911
(ln705542)^2=181.3525901334
705542/181.3525901334=3890.4434697129
2911/3890.4434697129=0.7482437472
【4(q23)^2】
D(746130)=12684
(ln746130)^2=182.8622016767
746130/182.8622016767=4080.2855546886
12684/4080.2855546886=3.1086059615

D(743044)=3214
(ln743044)^2=182.7501272560
743044/182.7501272560=4065.9014095193
3214/4065.9014095193=0.7904766192

D(744428)=3076
(ln744428)^2=182.8004432598
744428/182.8004432598=4072.3533637279
3076/4072.3533637279=0.7553372032
【4(q24)^2】
D(863940)=13626
(ln863940)^2=212.1476018498
863940/212.1476018498=4072.3533637287
13626/4072.3533637287=3.3459768303

D(850084)=3499
(ln850084)^2=186.4068787571
850084/186.4068787571=4560.3681884922
3499/4560.3681884922=0.7672626102

D(850118)=3443
(ln850118)^2=186.4079708761
850118/186.4079708761=4560.5238660371
3443/4560.5238660371=0.7549571280
【4(q25)^2】
D(1111110)=17629
(ln1111110)^2=193.7906236067
1111110/193.7906236067=5733.5591336710
17629/5733.5591336710=3.0747044879

D(1085764)=4343
(ln1085764)^2=193.1486904251
1085764/193.1486904251=5621.3893949286
4343/5621.3893949286=0.7725847998

D(1086728)=4244
(ln1086728)^2=193.1733586908
1086728/193.1733586908=5625.6618788694
4244/5625.6618788694=0.7544001206
【4(q26)^2】
D(1345890)=20104
(ln1345890)^2=199.1645208606
1345890/199.1645208606=6757.6795012703
20104/6757.6795012703=2.9749857175

D(1295044)=5060
(ln1295044)^2=198.0790305994
1295044/198.0790305994=6538.0166496227
5060/6538.0166496227=0.7739350129

D(1295642)=4938
(ln1295642)^2=198.0920254945
1295642/198.0920254945=6540.6065527659
4938/6540.6065527659=0.7549758513
【4(q27)^2】

D(1452990)=21618
(ln1452990)^2=201.3315253788
1452990/201.3315253788=7216.9025554554
21618/7216.9025554554=2.9954679080

D(1435204)=5492
(ln1435204)^2=200.9821560544
1435204/200.9821560544=7140.9523520662
5492/7140.9523520662=0.7690850925

D(1436924)=5381
(ln1436924)^2=201.0161171580
1436924/201.0161171580=7148.3024362199
5381/7148.3024362199=0.7527661355
【4(q28)^2】
D(1531530)=24044
(ln1531530)^2=202.8282347252
1531530/202.8282347252=7550.8718106973
24044/7550.8718106973=3.1842680690

D(1522756)=5857
(ln1522756)^2=202.6646187481
1522756/202.6646187481=7513.6746088507
5857/7513.6746088507=0.7795120637

D(1525532)=5648
(ln1525532)^2=202.7164796811
1525532/202.7164796811=7525.4463889658
5648/7525.4463889658=0.7505202626
【4(q29)^2】
D(1651650)=24111
(ln1651650)^2=204.9846596848
1651650/204.9846596848=8057.4322124383
24111/8057.4322124383=2.9923925345

D(1643524)=6210
(ln1643524)^2=204.8434562673
1643524/204.8434562673=8023.3170731867
6210/8023.3170731867=0.7739940904

D(1644044)=6062
(ln1644044)^2=204.8525116006
1644044/204.8525116006=8025.5008208315
6062/8025.5008208315=0.7553422690
【4(q30)^2】
D(1806420)=26597
(ln1806420)^2=207.5575442949
1806420/207.5575442949=8703.2249593078
26597/8703.2249593078=3.0559936259

D(1737124)=6507
(ln1737124)^2=206.4319938029
1737124/206.4319938029=8414.9940520295
6507/8414.9940520295=0.7732625787

D(1737686)=6348
(ln1737686)^2=206.4412890008
1737686/206.4412890008=8417.3374832651
6348/8417.3374832651=0.7541577147
【4(q31)^2】
D(2709630)=37527
(ln2709630)^2=219.4049023520
2709630/219.4049023520=12349.9063646847
37527/12349.9063646847=3.0386465202

D(2617924)=9154
(ln2617924)^2=218.3860977340
2617924/218.3860977340=11987.5945729325
9154/11987.5945729325=0.7636227555

D(2619728)=9023
(ln2619728)^2=218.4064579565
2619728/218.4064579565=11994.7368979436
9023/11994.7368979436=0.7522465959
【4(q32)^2】
D(2709630)=37527
(ln2709630)^2=219.4049023520
2709630/219.4049023520=12349.9063646847
37527/12349.9063646847=3.0386465202

D(2696164)=9361
(ln2696164)^2=219.2573350528
2696164/219.2573350528=12296.8018349340
9361/12296.8018349340=0.7612548470

D(2704766)=9251
(ln2704766)^2=219.3516792134
2704766/219.3516792134=12330.7284890608
9251/12330.7284890608=0.7502395344
【4(q33)^2】
D(2762760)=39092
(ln2762760)^2=219.9805333200
2762760/219.9805333200=12559.1112918209
39092/12559.1112918209=3.1126406233

D(2735716)=9551
(ln2735716)^2=219.6888308089
2735716/219.6888308089=12452.6858735923
9551/12452.6858735923=0.7669831309

D(2741798)=9390
(ln2741798)^2=219.7546662290
2741798/219.7546662290=12476.6315412063
9390/12476.6315412063=0.7526069812

小草

施承忠哥德巴赫偶数素数对数D(x)分段系数法

4(qk)^2≤x<4(qk+1)^2；D(2n)≈dk*2n/ln^2(2n)

d1=1.0701339985
d2=1.6966073953
d3=1.5003002109
d4=1.5490237640
d5=1.2742780425
d6=1.2245285542
d7=1.0788009896
d8=1.1498139954
d9=0.9308780418
d10=1.1166391978
d11=0.9753150060
d12=1.0672275196
d13=0.9810501357
d14=1.0658378014
d15=1.1936574232
d16=1.1070468417
d17=1.1748219094
d18=1.1102599444
d19=1.1742929922
d20=1.1226957192



比如D(100)=6，4(q2)^2=100,4(q3)^2=484.100=100<484,取密度系数
d2=1.6966073953.
100/(ln100)^2=4.7152924253
4.7152924253*1.6966073953=7.9999999998
4.7152924253*0.6601618158=3.1128560095
7.9999999998*1.3333333333=10.66666666613
3.1128560095*1.3333333333=4.1504746792

比如D(1000)=28，4(q3)^2=484,4(q4)^2=1156.484<1000<1156,取密度系数d3=1.5003002109
1000/(ln1000)^2=20.9568552235
20.9568552235*1.5003002109=31.4415743116
20.9568552235*0.6601618158=13.8349155978
31.4415743116*1.3333333333=41.9220990811
13.8349155978*1.3333333333=18.4465541299

比如D(10000)=127，4(q6)^2=6724,4(q7)^2=13924.
6724<10000<13924,取密度系数d6=1.2245285542
10000/(ln10000)^2=117.8823106323
117.8823106323*1.2245285542=144.3502554043
117.8823106323*0.6601618158=77.8214002377
144.3502554043*1.3333333333=192.4670072009
77.8214002377*1.3333333333=103.7618669810

比如D(100000)=810，4(q12)^2=75188,4(q13)^2=128164.
75188<100000<128164,取密度系数1.0672275196
100000/(ln100000)^2=754.4467880462
754.4467880462*1.0672275196=805.1663742767
754.4467880462*0.6601618158=498.0569615211
805.1663742767*1.3333333333=1073.5551656754
498.0569615211*1.3333333333=664.0759486782

比如D(10^6)=5402，4(q24)^2=850084,4(q25)^2=1085764.
850084<10^6<1085764,取密度系数d24=0.8172981975
10^6/(ln10^6)^2=5239.2138058788
5239.2138058788*0.8172981975=4281.9999998619
5239.2138058788*0.6601618158=3458.7288994534
4281.9999998619*1.3333333333=5709.3333330065
3458.7288994534*1.3333333333=4611.6385324892

比如D(10^7)=38807，4(q50)^2=8844676,4(q)^2=10329796.
8844676<10^7<10329796,取密度系数d50=0.7987595806
10^7/ln10^7=38492.1830636008
38492.1830636008*0.7987595806=30746.0000002602
38492.1830636008*0.6601618158=25411.0694653727
30746.0000002602*1.3333333333=40994.6666659887
25411.0694653727*1.3333333333=33881.4259529832

比如D(10^8)=291400，4(q126)^2=98684356,4(q127)^2=100360324.
98684356<10^8<100360324,取密度系数d126=0.9404091200
10^8/ln10^8=294705.7765806619
294705.7765806619*0.9404091200=277144.0000131369
294705.7765806619*0.6601618158=194553.5005942389
277144.0000131369*1.3333333333=369525.3333416111
194553.5005942389*1.3333333333=259404.6674525001

比如D(10^9)=2274205，4(q282)^2=990612676,
4(q283)^2=1009587076.
990612676<10^9<1009587076,取密度系数d282=0.7919290286
10^9/ln10^9=2328539.4692789145
2328539.4692789145*0.7919290286=1844037.9999628103
2328539.4692789145*0.6601618158=1537212.8442011365
1844037.9999628103*1.3333333333=2458717.3332222791
1537212.8442011365*1.3333333333=2049617.1255502749

比如D(10^10)=18200488，4(q705)^2=9996400324,
4(q706)^2=10008401764.
9996400324<10^9<10008401764,取密度系数d705=0.8201615936
10^10/ln10^10=18861169.7011628088
18861169.7011628088*0.8201615936=15469206.9992657250
18861169.7011628088*0.6601618158=12451424.0380315832
15469206.999265725038*1.3333333333=20625609.3318386598
12451424.038031583229*1.3333333333=16601898.7169603968

小草

施承忠哥德巴赫偶数素数对数D(x)真值分段系数法

d1=1.4268453314
d2=1.2724555465
d3=1.1054843659
d4=0.9466256335
d5=0.9214010461
d6=0.8202030882
d7=0.8565026038
d8=0.8526163101
d9=0.8535350591
d10=0.8374793984
d11=0.8108040411
d12=0.8026135729
d13=0.8148436221
d14=0.8148814464
d15=0.7841142055
d16=0.7867006100
d17=0.7963200124
d18=0.7862519980
d19=0.7838773800
d20=0.7829775785









比如D(100)=6，4(q2)^2=100,4(q3)^2=484.100=100<484,取密度系数
d2=1.2724555465.
100/(ln100)^2=4.7152924253
4.7152924253*1.2724555465=5.9999999999
4.7152924253*0.6601618158=3.1128560095
3.1128560095*1.3333333333=4.1504746792

比如D(1000)=28，4(q3)^2=484,4(q4)^2=1156.484<1000<1156,取密度系数d3=1.1054843659
1000/(ln1000)^2=20.9568552235
20.9568552235*1.1054843659=23.1674758080
20.9568552235*0.6601618158=13.8349155978
23.1674758080*1.3333333333=30.8899677432
13.8349155978*1.3333333333=18.4465541299

比如D(10000)=127，4(q6)^2=6724,4(q7)^2=13924.
6724<10000<13924,取密度系数d6=0.8202030882
10000/(ln10000)^2=117.8823106323
117.8823106323*0.8202030882=96.6874352248
117.8823106323*0.6601618158=77.8214002377
96.6874352248*1.3333333333=128.9165802965
77.8214002377*1.3333333333=103.7618669810

比如D(100000)=810，4(q12)^2=75188,4(q13)^2=128164.
75188<100000<128164,取密度系数d12=0.8026135729
100000/(ln100000)^2=754.4467880462
754.4467880462*0.8026135729=605.5292321167
754.4467880462*0.6601618158=498.0569615211
605.5292321167*1.3333333333=807.3723094687
498.0569615211*1.3333333333=664.0759486782

比如D(10^6)=5402，4(q24)^2=850084,4(q25)^2=1085764.
850084<10^6<1085764,取密度系数d24=0.6678482936
10^6/(ln10^6)^2=5239.2138058788
5239.2138058788*0.6678482936=3499.0000000617
5239.2138058788*0.6601618158=3458.7288994534
3499.0000000617*1.3333333333=4665.3333332990
3458.7288994534*1.3333333333=4611.6385324892

比如D(10^7)=38807，4(q50)^2=8844676,4(q)^2=10329796.
8844676<10^7<10329796,取密度系数d50=0.6845285952
10^7/ln10^7=38492.1830636008
38492.1830636008*0.6845285952=26348.9999987079
38492.1830636008*0.6601618158=25411.0694653727
26348.9999987079*1.3333333333=35131.9999973989
25411.0694653727*1.3333333333=33881.4259529832

比如D(10^8)=291400，4(q126)^2=98684356,4(q127)^2=100360324.
98684356<10^8<100360324,取密度系数d126=0.7327104426
10^8/ln10^8=294705.7765806619
294705.7765806619*0.7327104426=215933.9999951935
294705.7765806619*0.6601618158=194553.5005942389
215933.9999951935*1.3333333333=287911.9999863935
194553.5005942389*1.3333333333=259404.6674525001

小草

【3】1-(1/4)=0.75
【5】1-(1/16)=0.9375=0.703125
【7】1-(1/36)=0.9722222222=0.6835937500
【11】1-(1/100)=0.99=0.6767578125
【13】1-(1/144)=0.9930555556=0.6720581055
【17】1-(1/256)=0.99609375=0.6694328785
【19】1-(1/324)=0.99691358=0.6673667275=1.334733455